292 Mr. G. H. Livens on the Electron 



The new component along the direction of the .r-axis is 

 therefore 



f' = £-2£cos 2 <9 + Vw 2 -Psin2(9 cosi/r, 



where (f, 77, f) are used for the components of w, and i/r is 

 the azimuth angle between the plane of the motion of the 

 electron and the plane through the polar axis parallel to the 

 A'-axis. Similar expressions hold for 1/ and f. 



The question of the frequency of the collision next arises. 

 The law of force assumed is such that the electrons are in 

 reality influenced by the atoms at all distances, although of 

 course the influence may only become perceptible when the 

 electrons are near enough to the atoms. We may therefore 

 firstly suppose that in the general case the influence is 

 perceptible when the electrons come within a distance a 

 from the centre of the atoms ; this will give us an approxi- 

 mation which is the better the greater is the value of a. To 

 obtain the actual exact result we may then in our final 

 formulae pass to the limit when a is infinite. 



With this definition we see that the probability of an 

 electron with its velocity components in the specified range, 

 and describing an orbit for which the elements p and ^ lie 

 within the limits (p, ^) and (p + dp, yjr + dyp-} colliding with 

 an atom in the next succeeding small interval of time dt, is 

 equal to the probability of finding an atom in a certain small 

 cylinder of height udt and base pdpd-^r, which is 



nupdpdtydt, 



wherein n denotes the number of atoms per unit volume in 

 the metal. 



The total number of the electrons per unit volume in the 

 group specified as having their velocity components between 

 (f, 7), f) and (f + df, r\ + dt), f+df), which collide during 

 the same small time dt, is therefore 



i 



c 



nu dN dt dyjrp dp 

 = Tr'a 2 nudl$dt, 



wherein 



dK=fd£d v d£ 



Moreover, the velocity of each of these electrons has the 

 same £ component before collision, and therefore the average 



